Non-Local Tug-of-War and the Infinity Fractional Laplacian

Motivated by the classical ``tug-of-war'' game,
we consider a ``non-local'' version of the game which goes as follows: at every step two players pick respectively a direction
and then, instead of flipping a coin in order to decide which direction to choose and then moving of a fixed amount $\epsilon>0$ (as is done in the classical case),
it is a $s$-stable Levy process which chooses at the same time both the direction and the distance to travel. Starting from this game,
we heuristically we derive a deterministic non-local integro-differential equation that we call ``infinity fractional Laplacian''.
We study existence, uniqueness, and regularity, both for the Dirichlet problem and for a double obstacle problem, both problems having a natural interpretation as ``tug-of-war'' games.